1 Integers and powers
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1 1 Integers and powers 1.1 Integers and place value An integer is any positive or negative whole number. Zero is also an integer. The value of a digit in a number depends on its position in the number. This is its place value. Example 1 (a) Which of the following numbers are integers? 30, 0.3, 3, 0, 3.3 (b) Write them in order of size, largest first. (a) 30, 3 and 0 are integers. (b) 30, 0.3, 0, 3, and 3.3 are not whole numbers. Example 2 (a) Write in words. (b) Write five hundred and forty thousand two hundred and sixteen in figures. (a) Two million, four hundred and sixty-seven thousand, three hundred and eighty-one (b) is millions thousands hundreds tens units Exercise 1A 1 From each list of numbers write down the integers in order of size, starting with the largest. (a) 6.3, 63, 6.3, 63, 0 (b) 62,.62, 620, 0.62 (c) 1 million, 0.01, 2, , Write these numbers in words. (a) 432 (b) 8200 (c) 6370 (d) (e) (f) (g) (h)
2 2 Chapter 1 Integers and powers 3 Write these numbers in digits. (a) four hundred and sixty-three (b) fifteen thousand and twenty-seven (c) one hundred and sixteen thousand two hundred and twenty-five (d) three hundred and five thousand one hundred and one (e) two million, three hundred and twenty-seven thousand and thirty-five (f) half a million 1.2 Calculating with negative numbers You can use negative numbers to describe quantities such as temperatures less than 0 C. Subtracting a positive number is the same as adding the negative number. Subtracting a negative number is the same as adding the positive number. Adding a negative number is the same as subtracting the positive number. This table shows the signs you get when you multiply or divide two numbers. Negative number positive number negative answer. Example 3 Work out (a) 2 3 (c) 4 2 (b) 3 2 (d) 3 1 (a) is the same as 2 3. Start at 2 and go down 3 to get to
3 1.2 Calculating with negative numbers 3 (b) is the same as 3 2. Start at 3 and go up 2 to get to (c) is the same as 4 2. Start at 4 and go down 2 to get to (d) Start at 3 and go up 1 to get to Example 4 Work out (a) 1 3 (b) 8 2 (c) 16 3 (d) 10 (a) (c) (b) (d) 10 2 Exercise 1B 1 Work out (a) 4 3 (b) 9 (c) 8 2 (d) 4 (e) 7 6 (f) 2 4 (g) 6 8 (h) Work out (a) 3 8 (b) 3 (c) 24 3 (d) (e) 8 (f) 48 8 (g) 6 (h) 0 3 A diver dives to a depth of 27 metres. A second diver dives to a depth of 16 metres. What is the difference in the depths of the dives? 4 The temperature at the Arctic Circle is recorded as 18 C one night. The following day it rises by 6 C. What is the temperature during the day?
4 4 Chapter 1 Integers and powers Copy and complete the following tables. (a) 1st number nd 30 number (b) 1st number nd 4 number 8 1 (c) 1st number (d) 1st number nd 2 2nd 2 12 number 1 number Rounding to one significant figure The first significant figure is the first non-zero digit in a number, counting from the left. To write a number to one significant figure (1 s.f.), look at the place value of the first significant figure and round to this place value. To estimate the answer to a calculation, round all numbers to one significant figure and do the simpler calculation. Example Estimate the answer to to 1 s.f to 1 s.f to 1 s.f. 30 This digit is greater than so round up. This digit is less than so round down. This digit is so round up. First round all the numbers to one significant figure. An estimate is Example Estimate the answers to these questions by rounding all the numbers to 1 s.f. (a) (b) (c) (a) becomes as an estimate. (b) becomes as an estimate. (c) becomes as an estimate. 10
5 1.4 Rounding to a number of significant figures Exercise 1C 1 Write these numbers to 1 significant figure. (a) 36 (b) 237 (c) 84 (d) 3.21 (e) 16.8 (f) 800 (g) 219 (h) For each of the following calculations (i) write down a calculation that can be used to estimate the answer (ii) work out an estimated answer. (a) (b) (c) (d) (e) (f) spectators watched a pop concert. They paid 18 for a ticket. Estimate the total income from the ticket sales. 4 There are 1.76 pints in 1 litre. Estimate how many pints there are in 19 litres. 1.4 Rounding to a number of significant figures You can round numbers to a given number of significant figures (s.f.). Example 7 Round to (a) 1 s.f. (b) 2 s.f. (c) 3 s.f. (d) 4 s.f. (e) s.f. 1 s.f. 2 s.f. 3 s.f. 4 s.f. s.f. 6 s.f. You need zeros to show the place value of the 7. (a) (1 s.f.) (b) (2 s.f.) (c) (3 s.f.) (d) (4 s.f.) (e) ( s.f.) You need a zero here to show significant figures.
6 6 Chapter 1 Integers and powers Exercise 1D 1 Round these numbers to 2 s.f. (a) 4.6 (b) 437 (c) (d) (e) (f) 0.09 In the first significant figure is 2. 2 Round these numbers to 3 s.f. (a) 94.3 (b) (c) (d) (e) (f) Round these numbers to the number of significant figures given in the brackets. (a) 4892 (2 s.f.) (b) (1 s.f.) (c) (2 s.f.) (d) (3 s.f.) (e) (3 s.f.) (f) 4999 (3 s.f.) 4 Work out an estimate for (a) (b) (c) (d) (g) (e) (h) (f) ( ) 407 Round each number to 1 s.f. 1. Squares, square roots, cubes and cube roots You need to memorise all the square numbers from 2 2 to 1 2 and the corresponding square roots. If x x A, then x is the square root of A, written A. For example, , so 4 is the square root of 16, written 16 4 You need to memorise the cubes of 2, 3, 4, and 10 and the corresponding cube roots. To find the cube of any number, multiply the number by itself, then multiply by the number again. For example If y y y A, then y is the cube root of A, written 3 A. For example, , so 2 is the cube root of 8, written You can use the square root function key on a calculator to find the square root of a positive number. You can use the cube root function key on a calculator to find the cube root of any positive or negative number. Notice that , so 4 is also a square root of x 2x 2 8 So For example (2 d.p.) For example (2 d. p.)
7 1.6 Index notation 7 Exercise 1E 1 Write down from memory (a) 3 2 (b) 4 3 (c) 7 2 (d) 8 2 (e) 10 2 (f) 2 2 (g) 6 2 (h) 3 3 (i) 10 3 (j) 4 2 (k) 3 (l) 12 2 (m) 13 2 (n) 2 3 (o) 2 (p) 9 2 (q) 11 2 (r) 14 2 (s) Write down from memory (a) 64 (b) 9 (c) 2 (d) 144 (e) 169 (f) 22 (g) 4 (h) 100 (i) 36 (j) 121 (k) 49 (l) 81 (m) 16 (n) 196 (o) 3 8 (p) 3 12 (q) 3 27 (r) 3 64 (s) Estimate, to the nearest whole number, the value of (a) 0 (b) 38 (c) 128 (d) 69 (e) 24 4 Use your calculator to work out (a) 77 (b) 3 96 (c) (d) ( 1.3) 3 (e) (f) 23 (g) ( 71.7) 2 (h) (i) Give your answers to 3 significant figures. 1.6 Index notation The 2 in 7 2 is called an index or a power. It tells you how many times the base number (7, here) must be multiplied by itself. Example 8 Rewrite these expressions using index notation. (a) (b) (c) (d) 3 (a) 3 3 (b) (c) 2 (d) 3 1 Example 9 Evaluate (a) 4 3 (b) (c) 4 (a) (b) (c) 4 62
8 8 Chapter 1 Integers and powers Example 10 Find the value of x when (a) 6 x 36 (b) 2 x 32 (c) 4 x 26 (a) so and x 2. (b) so 2 32 and x. (c) so and x 4. Exercise 1F 1 Rewrite these expressions using index notation. (a) (b) (c) (d) (e) (f) (g) Evaluate (a) 2 4 (b) 3 (c) 6 3 (d) 7 4 (e) 8 3 (f) (g) (h) (i) (j) Find x when (a) x 12 (b) 3 x 81 (c) 2 x 64 (d) 10 x (e) 9 x 81 (f) 3 x 27 (g) 2 x 16 (h) 7 x Index laws To multiply powers of the same number, add the indices. In general: x n x m x n m To divide powers of the same number, subtract the indices. In general: x n x m x n m Indices is the plural of index. Example 11 Simplify (a) (b) (c) (d) (e) (f) x3 x 2 (a) (b) (c) (d) (e) (f) x 3 x 2 x 3 2 x
9 1.8 BIDMAS 9 Exercise 1G Simplify x4 x 3 18 y 4 y 2 a 2 a 4 a BIDMAS 20 z 4 z z 2 BIDMAS is a made-up word to help you remember the order of operations. B I D M A S Brackets Indices Divide Multiply Add Subtract (powers and roots) When operations are the same, do them in the order they appear. Example 12 Work out (a) (b) (a) (c) ( ) 2 ((2 2) (3 3)) 2 (4 9) (b) (c) ( ) This line acts as a bracket. 4 2 (48 16)
10 10 Chapter 1 Integers and powers Exercise 1H 1 Work out (a) 4 7 (b) (c) (6 2) 3 (d) ( 3) (6 4) (e) 9 3 (f) (2 7) 2 2 (g) ( 4) (h) (i) ( 20) 3 2 (j) (k) ( 4) 2 (3 ) 2 (l) (10 6) 2 Make these expressions correct by replacing the * with,, or. Use brackets if you need to. (a) 3 * 4 * 27 (b) 2 * 3 * 2 * 3 2 (c) 6 * 7 * 8 * 9 1 (d) 10 * 9 * 8 * 7 34 (e) 3 * 3 * 3 0 (f) 3 * 3 * Reciprocals The reciprocal of n is 1 n or 1 n. Multiplying a number by its reciprocal gives 1. Zero has no reciprocal because you cannot divide by zero. Example 13 Find the reciprocal of (a) 6 (b) 0.2 (c) 2_ (a) Check: reciprocal of (c) (b) Check: Check: Exercise 1I 1 Find the reciprocal of (a) 4 (b) 3 (c) 2 (d) 7 (e) 9 (f) 20
11 1.10 LCM, HCF and prime factor decomposition 11 2 Find the reciprocal of (a) 0.1 (b) 0. (c) 0.2 (d) 0.4 (e) 0.3 (f) Find the reciprocal, in its simplest form, of (a) 3_ (b) 2_ 7 (c) 1_ (d) 1_ 4 8 (e) 3 16 (f) _ 8 4 Find the reciprocal of (a) x (b) y (c) 1 x (d) y LCM, HCF and prime factor decomposition A factor of a number is a whole number that divides exactly into it. The factors of a number include 1 and the number itself. Multiples of a number are the results of multiplying the number by the positive whole numbers. A prime number is a whole number greater than 1 that only has two factors: itself and 1. A number written as a product of prime numbers is written in prime factor form. The highest common factor (HCF) of two numbers is the highest factor common to both of them. The lowest common multiple (LCM) of two numbers is the lowest multiple common to both of them. Example 14 (a) Write 36 in prime factor form. (b) Find the highest common factor (HCF) of 36 and 12. (c) Find the lowest common multiple (LCM) of 3 and 4. Method 1 Method 2 (a) Using a factor tree. which can be simplified to (b) HCF of 24 and 36 is (c) 3: 3, 6, 9, 12, 1 4: 4, 8, 12, 16 LCM of 3 and 4 is Write each number in prime factor form. Pick out the factors common to both numbers. Write a list of multiples for each number. The LCM is the lowest number that appears in both lists.
12 12 Chapter 1 Integers and powers Exercise 1J 1 Write down all the factors of (a) 48 (b) 360 (c) 29 (d) 100 (e) 71 (f) 64 2 Write down the numbers in question 1 that are prime numbers. 3 Write down the first five multiples of (a) 4 (b) 7 (c) 11 (d) 20 4 Write these numbers in prime factor form, using index notation. (a) 0 (b) 72 (c) 40 (d) 840 Find the HCF of (a) 9 and 1 (b) 4 and 14 (c) 12 and 20 (d) 6, 1 and 21 (e) 8, 24 and 36 6 Find the LCM of (a) 6 and 8 (b) and 7 (c) 4 and 6 (d) 2, 3 and 4 (e), 6 and 10 7 Given that a and b write down (a) the LCM of a and b (b) the HCF of a and b. Mixed exercise 1 1 A list of numbers is written below three hundred and two thousand four hundred and twelve From the list (a) write down the integers (b) write down the first number in words (c) write the fourth number in figures. 2 Work out (a) 6 3 (b) 7 3 (c) 8 4 (d) 3 3 Work out (a) 8 2 (b) 10 (c) 6 3 (d) 7
13 Mixed exercise For each of the following calculations (i) write down a calculation that could be used to estimate the answer (ii) write down the estimated answer. (a) (b) (c) Estimate the cost of 27 books at 19.9 each. 6 Round each number to the number of significant figures given in the brackets. (a) 37.2 (2 s.f.) (b) 473 (1 s.f.) (c) (2 s.f.) (d) (3 s.f.) (e) ( s.f.) 7 Write down from memory (a) 4 2 (b) 12 2 (c) 8 2 (d) 13 2 (e) 49 (f) 196 (g) 22 (h) 4 3 (i) 3 (j) Rewrite these expressions using index notation. (a) (b) (c) (d) Evaluate (a) 8 3 (b) 10 4 (c) 4 (d) (e) Find x when (a) 9 x 81 (b) 2 x 16 (c) 10 x Here are some numbers From the numbers in the cloud write down (a) the square numbers (b) the cube numbers (c) the prime numbers (d) factors of 27 (e) multiples of Simplify (a) (b) 3 2 (c) (d) (e) (f) (g) (h) (i) 3 6 4
14 14 Chapter 1 Integers and powers 13 Write down the value of (a) 2. 2 (b) the square of 10.1 (c) 6.4 cubed (d) (e) 38 (f) (g) 6.3 squared (h) the cube of 2.4 (i) the positive square root of.76 (j) Use BIDMAS to work out the value of (a) (7 3) ( 3) (b) (c) (2 ) 2 (9 2) (d) (3 9) 1 Write in prime factor form (a) 180 (b) 196 (c) Find the highest common factor (HCF) of (a) 12 and 18 (b) 42 and 24 (c) 6, 12 and 1 17 Find the lowest common multiple (LCM) of (a) 4 and (b) 6 and 8 (c) 2, 6 and 8 18 Find the reciprocal of (a) 9 (b) 6 (c) 0. (d) 0.62 (e) 4 (f) 3 (g) x (h) 1 7 y Summary of key points 1 An integer is any positive or negative whole number. Zero is also an integer. 2 The value of a digit in a number depends on its position in the number. This is its place value. 3 You can use negative numbers to describe quantities such as temperatures less than 0 C. 4 Subtracting a positive number is the same as adding the negative number. Subtracting a negative number is the same as adding the positive number. 6 Adding a negative number is the same as subtracting the positive number. 7 This table shows the signs you get when you multiply or divide two numbers. 8 The first significant figure is the first non-zero digit in a number, counting from the left. 9 To write a number to one significant figure (1 s.f.), look at the place value of the first significant figure and round to this place value.
15 Summary of key points 1 10 To estimate the answer to a calculation, round all numbers to one significant figure and do the simpler calculation. 11 You can round numbers to a given number of significant figures (s.f.). 12 You need to memorise all the square numbers from 2 2 to 1 2 and the corresponding square roots. 13 If x x A, then x is the square root of A, written A. 14 You need to memorise the cubes of 2, 3, 4, and 10 and ther corresponding cube roots. 1 To find the cube of any number, multiply the number by itself, then multiply by the number again. For example If y y y A, then y is the cube root of A, written 3 A. For example, , so 2 is the cube root of 8, written You can use the square root function key on a calculator to find the square root of a positive number. 18 You can use the cube root function key on a calculator to find the cube root of any positive or negative number. 19 The 2 in 7 2 is called an index or a power. It tells you how many times the base number (7, here) must be multiplied by itself. 20 To multiply powers of the same number, add the indices. In general: x n x m x n m 21 To divide powers of the same number, subtract the indices. In general: x n x m x n m 22 BIDMAS is a made-up word to help you remember the order of operations. B I D M A S Brackets Indices Divide Multiply Add Subtract (powers and roots) 23 When operations are the same, do them in the order they appear. 24 The reciprocal of n is 1 n or 1 n. 2 Multiplying a number by its reciprocal gives Zero has no reciprocal because you cannot divide by zero. 27 A factor of a number is a whole number that divides exactly into it. The factors of a number include 1 and the number itself. 28 Multiples of a number are the results of multiplying the number by the positive whole numbers. 29 A prime number is a whole number greater than 1 that only has two factors: itself and A number written as a product of prime numbers is written in prime factor form. 31 The highest common factor (HCF) of two numbers is the highest factor common to both of them. 32 The lowest common multiple (LCM) of two numbers is the lowest multiple common to both of them.
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